Lower Bounds for Enumerative Counts of Positive-Genus Real Curves

TL;DR

This paper establishes lower bounds for counting real algebraic curves of positive genus in symplectic threefolds by transforming Gromov-Witten invariants into signed curve counts, with implications for Hodge integrals.

Contribution

It introduces a method to convert positive-genus real Gromov-Witten invariants into signed counts, providing new lower bounds for real curve enumeration.

Findings

Integer invariants serve as lower bounds for real curve counts

Transformation of Gromov-Witten invariants into signed counts

Implications for Hodge integrals and related conjectures

Abstract

We transform the positive-genus real Gromov-Witten invariants of many real-orientable symplectic threefolds into signed counts of curves. These integer invariants provide lower bounds for counts of real curves of a given genus that pass through conjugate pairs of constraints. We conclude with some implications and related conjectures for one- and two-partition Hodge integrals.